Are there any discount-factors based on branching factors?
First, an important note on any form of discounting: adding a discount factor can change what the optimal policy is. The optimal policy when a discount factor is present can be different from the optimal policy in the case where a discount factor is absent. This means that "artificially" adding a discount factor is harmful if we expect to be capable of learning / finding an optimal policy. Generally, we don't expect to be capable of doing that though, except for the case where we have an infinite amount of processing time (which is never in practice). In that other answer which you linked to in your question, I describe that it can still be useful, that it may help to find a good (not optimal, just good) policy more quickly, but it does not come "for free". 



I'm not aware of any research evaluating ideas like the one in your question. I am not 100% sure why, I suspect it could be an interesting idea in some situations, but would have to be investigated carefully due to my point above; if not evaluated properly, it could also unexpectedly be harmful.

One thing to note is that the use of discounting factors $\gamma &lt; 1$ is extremely common (basically ubiquitous) in Reinforcement Learning (RL) literature, but rather rare in literature on tree search algorithms like MCTS (though not non-existant; for example, it's used in the original UCT paper from 2006). For the concept of "branching factors", we have the opposite; in RL literature it is very common to consistently have the same action space regardless of states ("constant branching factor"), whereas this is very uncommon in literature on tree search algorithms. So, the combination of discount factors + branching factors is actually somewhat rare in existing literature (which of course doesn't mean that the idea couldn't work or be relevant, it just might explain why the idea doesn't appear to have been properly investigated yet).

One important concern I do have with your idea is that it seems like it could be somewhat of an "anti-heuristic" in some situations (with which I mean, a heuristic that is detrimental to performance). In many games, it is advantageous to be in game states where you have many available moves (a large branching factor), this can mean that you are in a strong position. Consider, for example, chess, where a player who is in a convincingly winning position likely has more moves available than their opponent. I suspect your idea, when applied to chess, would simply promote an aggressive playing style where both players capture as many pieces as possible in an effort to quickly reduce the branching factors across the entire tree.




  If not, I'd welcome thoughts on how this might be applied.
  
  (I might divide 1 by the number of branches and add that value to the goodness of a given state, but that's a "value-add" as opposed to a discount.)


Such an additive change would be more closely related to the idea of reward shaping, rather than discounting (which is again a thing that, if not done carefully, can significantly alter the task that you're optimizing for). Intuitively I also suspect it might not do anything at all since you'd always be adding the same constant value regardless of which move you took (regardless of which move you took, your parent state will always have had the same branching factor). I might be missing out on some details here, but I think you'd have to have a multiplicative effect on your regular observed rewards.

I suppose one example that could be worth a try would be something like maximizing the following return;

$$\sum_{t = 0}^{T} \gamma^t \beta^{b(S_{t}) - 1} R_{t + 1},$$

where:


$0 \leq \gamma \leq 1$ is the normal time-based discount factor (can set it to $1$ if you like)
$0 \leq \beta \leq 1$ is a new branching-factor-based discount factor
$b(S_t)$ is the branching factor (the number of available moves) in state $S_t$
$R_{t + 1}$ is the immediate reward for transitioning from state $S_t$ to $S_{t + 1}$


I put the $-1$ in the power of $\beta$ because I suspect you wouldn't want to do any discounting in situations where only one move is available. Intuitively, I do suspect this $\beta$ would require very careful tuning though. It is already quite common to choose $\gamma = 0.9$ or $\gamma = 0.99$, with $\beta$ you may want to stay even closer to $1$.